On a Geometric Property of the Set of Invariant Means on a Group

Chou, Ching

doi:10.2307/2038270

Cited by 3 publications

(4 citation statements)

References 4 publications

(6 reference statements)

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“…Since a set is central in N if and only if it is a member of an idempotent in some minimal left ideal L of βN, and since there are 2 c minimal left ideals of βN [8], the assertion that a set is a member of an idempotent in every minimal left ideal L of βN is considerably stronger than the assertion that it is central. To see this, let p be any idempotent in K(βN) and let L be a minimal left ideal of βN for which p / ∈ L. Since L is closed, by [15, Corollary 2.6], there is a subset A of N for which A ∈ p and L ∩ c A = ∅.…”

Section: Strongly and Very Strongly Central Setsmentioning

confidence: 98%

Strongly central sets and sets of polynomial returns mod 1

Bergelson

Hindman

Strauss

2012

Proc. Amer. Math. Soc.

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Abstract. Central sets in N were introduced by Furstenberg and are known to have substantial combinatorial structure. For example, any central set contains arbitrarily long arithmetic progressions, all finite sums of distinct terms of an infinite sequence, and solutions to all partition regular systems of homogeneous linear equations. We introduce here the notions of strongly central and very strongly central, which as the names suggest are strictly stronger than the notion of central. They are also strictly stronger than syndetic, which in the case of N means that gaps are bounded.Given. Kronecker's Theorem says that if 1, α 1 , α 2 , . . . , αv are linearly independent over Q and U is a nonempty open subset of (− ) v , then {x ∈ N : (w(α 1 x), . . . , w(αvx)) ∈ U } is nonempty and Weyl showed that this set has positive density. We show here that if 0 is in the closure of U , then this set is strongly central. More generally, let P 1 , P 2 , . . . , Pv be real polynomials with zero constant term. We show that {x ∈ N : (w (P 1 (x)), . . . , w(Pv(x))) ∈ U } is nonempty for every open U with 0 ∈ c U if and only if it is very strongly central for every such U and we show that these conclusions hold if and only if any nontrivial rational linear combination of P 1 , P 2 , . . . , Pv has at least one irrational coefficient.

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Section: Strongly and Very Strongly Central Setsmentioning

confidence: 98%

Strongly central sets and sets of polynomial returns mod 1

Bergelson

Hindman

Strauss

2012

Proc. Amer. Math. Soc.

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“…It is natural to ask how many exposed points (with respect to the w*-topology) ML(S) has. Chou 488 ZHUOCHENG YANG [1] proved that if G is a countable infinite amenable group, then ML(G) has no exposed points. Later, Granirer [4] studied intensively the existence of exposed points of subsets of ML(S) for a countable left amenable semigroup S. In particular, he proved [4,Cor.…”

Section: Ml(s)mentioning

confidence: 99%

“…Let M(S,K) denote the set of all μ e ML(S) with its support contained in K (see [1] for the definitions). Chou [1] proved that if G is a countably infinite amenable group, then M(G,K) has no exposed points. He asked whether this holds for any infinite amenable group.…”

Section: For Any Left Amenable Semigroup S Ml(s) Has Exposed Points mentioning

confidence: 99%

Exposed points of left invariant means

Yang¹

1986

Pacific J. Math.

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If S is a left amenable semigroup, let ML(S) be the set of left invariant means on m(S), the space of bounded real-valued functions on S. We prove in this paper that a left invariant mean on m(S) is an exposed point of ML(S) if and only if it is the arithmetic average on a minimal finite left ideal of S. In particular, ML(S) has no exposed point when S is an infinite group. We also prove that if ML(S) has an exposed point, then it is the w*-closed convex hull of all its exposed points. This gives another proof of the Granirer-Klawe theorem on the dimension of ML(S).

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“…It is natural to ask how many exposed points LIM has. Granirer [4] studied intensively the existence of exposed points of LIM for a countable amenable semigroup (also see Chou [1]). In particular, he proved by using very general theorems that LIM has exposed points if and only if G has finite left ideals for a countable amenable semigroup G [4, Corollary 4.1].…”

Section: Introduction and Notationsmentioning

confidence: 99%

The exposed points of the set of invariant means

Miao¹

1995

Trans. Amer. Math. Soc.

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Let G G be a σ \sigma -compact infinite locally compact group, and let L I M LIM be the set of left invariant means on L ∞ ( G ) {L^\infty }(G) . We prove in this paper that if G G is amenable as a discrete group, then L I M LIM has no exposed points. We also give another proof of the Granirer theorem that the set L I M ( X , G ) LIM(X,G) of G G -invariant means on L ∞ ( X , β , p ) {L^\infty }(X,\beta ,p) has no exposed points, where G G is an amenable countable group acting ergodically as measure-preserving transformations on a nonatomic probability space ( X , β , p ) (X,\beta ,p) .

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On a Geometric Property of the Set of Invariant Means on a Group

Cited by 3 publications

References 4 publications

Strongly central sets and sets of polynomial returns mod 1

Strongly central sets and sets of polynomial returns mod 1

Exposed points of left invariant means

The exposed points of the set of invariant means

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